| L(s) = 1 | + 2-s − 2.16·3-s + 4-s − 2.63·5-s − 2.16·6-s − 3.58·7-s + 8-s + 1.69·9-s − 2.63·10-s − 2.72·11-s − 2.16·12-s − 3.55·13-s − 3.58·14-s + 5.70·15-s + 16-s + 5.45·17-s + 1.69·18-s + 0.879·19-s − 2.63·20-s + 7.77·21-s − 2.72·22-s + 8.64·23-s − 2.16·24-s + 1.93·25-s − 3.55·26-s + 2.82·27-s − 3.58·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.25·3-s + 0.5·4-s − 1.17·5-s − 0.884·6-s − 1.35·7-s + 0.353·8-s + 0.565·9-s − 0.832·10-s − 0.820·11-s − 0.625·12-s − 0.987·13-s − 0.958·14-s + 1.47·15-s + 0.250·16-s + 1.32·17-s + 0.399·18-s + 0.201·19-s − 0.588·20-s + 1.69·21-s − 0.579·22-s + 1.80·23-s − 0.442·24-s + 0.386·25-s − 0.698·26-s + 0.544·27-s − 0.677·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 + 2.63T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 + 3.55T + 13T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 - 0.879T + 19T^{2} \) |
| 23 | \( 1 - 8.64T + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 4.17T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 6.00T + 53T^{2} \) |
| 59 | \( 1 - 1.97T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 16.9T + 73T^{2} \) |
| 79 | \( 1 - 2.73T + 79T^{2} \) |
| 83 | \( 1 - 2.17T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 0.589T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.46629719434878599997817533773, −6.79032460140924061387340498329, −5.81111824272100659329970502079, −5.50642959439123802453047298049, −4.79860170017788945027274133633, −3.93887294097810429792980006174, −3.23370233072828766519492929586, −2.60433118177813536361126900338, −0.859673557686044058842885918055, 0,
0.859673557686044058842885918055, 2.60433118177813536361126900338, 3.23370233072828766519492929586, 3.93887294097810429792980006174, 4.79860170017788945027274133633, 5.50642959439123802453047298049, 5.81111824272100659329970502079, 6.79032460140924061387340498329, 7.46629719434878599997817533773
